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# Beta plane approximation

Physics & Chemistry:

# Beta plane approximation

This article has been reviewed by the following Topic Editor: C Michael Hogan

The beta plane approximation, in oceanography and general fluid dynamics, is a simplified coordinate system for the equations of motion where the variation of the Coriolis parameter f with latitude is approximated by

= f0 + βy

where f0 is the value of f at the mid-latitude of the region and β the latitudinal gradient of f at that same latitude. This formalism is used to investigate both equatorial and mid-latitude phenomena (for which there are slightly different beta plane approximations) where f varies significantly over a few tens of degrees latitude. The beta plane approximation allows considerable simplification of the governing equations and therefore the use of analytical investigation methods. See Gill [1982].

The beta plane equations are obtained by introducing a background stratification into the shallow water equations, expanding them around a reference latitude θ0 with respect to ε~ θθ0, and keeping terms up to first order in ". This approximation introduces the horizonal coordinates

x = r0 cos θ0(φφ0)

y = r0(θθ0)

and expands the Coriolis parameter as

f = f0 + β0y + • • •

where β0 is the beta paramter at the reference latitude. The resulting equations (after Muller [1995])

are:

Image:Img62.png + Image:Img64.png - Image:Img66.png = Image:Img67.png

Image:Img68.png   +  Image:Img69.png   +  Image:Img70.png

0  =  Image:Img72.png

Image:Img73.png   =  Image:Img74.png

Image:Img75.png   +  Image:Img76.png

where (u, v,w) are the velocity components in the (x, y, z) directions, r0 is the mean radius of the Earth, θ0 is the reference latitude, f0 = 2Ωsin θ0 is the Coriolis parameter at the reference latitude, Image:Img81.png is the beta parameter at the reference latitude, ρ* is a constant reference density, δp and δρ are motionally induced deviations from prescribed background fields, and N is the buoyancy frequency.

 This article is written at a definitional level only. Authors wishing to expand this entry are inivited to expand the present treatment, which additions will be peer reviewed prior to publication of any expansion.

## Citation

Steve Baum (Lead Author);C Michael Hogan (Topic Editor) "Beta plane approximation". In: Encyclopedia of Earth. Eds. Cutler J. Cleveland (Washington, D.C.: Environmental Information Coalition, National Council for Science and the Environment). [First published in the Encyclopedia of Earth March 30, 2010; Last revised Date December 10, 2011; Retrieved May 22, 2013 <http://www.eoearth.org/article/Beta_plane_approximateion?topic=49557>

## The Author

Assistant Research Scientist, Physical Section Department of Oceanography Texas A&M University   ... (Full Bio)